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Contents
Preface v
I. Mathematical Bases
1. Manifolds, Flows, Lie Groups and Lie Algebras 3
1.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Manifolds and Diffeomorphisms . . . . . . . . . . . . . . . . 4
1.3 Flows and Vector Fields . . . . . . . . . . . . . . . . . . . . 8
1.3.1 A steady flow and its velocity field . . . . . . . . . . 8
1.3.2 Tangent vector and differential operator . . . . . . . 10
1.3.3 Tangent space . . . . . . . . . . . . . . . . . . . . . . 11
1.3.4 Time-dependent (unsteady) velocity field . . . . . . . 12
1.4 Dynamical Trajectory . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 Fiber bundle (tangent bundle) . . . . . . . . . . . . . 13
1.4.2 Lagrangian and Hamiltonian. . . . . . . . . . . . . . 14
1.4.3 Legendre transformation . . . . . . . . . . . . . . . . 16
1.5 Differential and Inner Product . . . . . . . . . . . . . . . . 18
1.5.1 Covector (1-form) . . . . . . . . . . . . . . . . . . . . 18
1.5.2 Inner (scalar) product . . . . . . . . . . . . . . . . . 20
1.6 Mapping of Vectors and Covectors . . . . . . . . . . . . . . 21
1.6.1 Push-forward transformation . . . . . . . . . . . . . 21
1.6.2 Pull-back transformation . . . . . . . . . . . . . . . . 23
1.6.3 Coordinate transformation . . . . . . . . . . . . . . . 24
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1.7 Lie Group and Invariant Vector Fields . . . . . . . . . . . . 25
1.8 Lie Algebra and Lie Derivative . . . . . . . . . . . . . . . . 27
1.8.1 Lie algebra, adjoint operator and Lie bracket . . . . 27
1.8.2 An example of the rotation group SO(3) . . . . . . . 29
1.8.3 Lie derivative and Lagrange derivative . . . . . . . . 30
1.9 Diffeomorphisms of a Circle S1 . . . . . . . . . . . . . . . . 35
1.10Transformation of Tensors and Invariance . . . . . . . . . . 36
1.10.1 Transformations of vectors and metric tensors . . . . 36
1.10.2 Covariant tensors . . . . . . . . . . . . . . . . . . . . 38
1.10.3 Mixed tensors . . . . . . . . . . . . . . . . . . . . . . 39
1.10.4 Contravariant tensors . . . . . . . . . . . . . . . . . . 42
2. Geometry of Surfaces in R3 45
2.1 First Fundamental Form . . . . . . . . . . . . . . . . . . . . 45
2.2 Second Fundamental Form. . . . . . . . . . . . . . . . . . . 50
2.3 Gauss’s Surface Equation and an Induced Connection . . . 52
2.4 Gauss–Mainardi–Codazzi Equation and Integrability . . . . 54
2.5 Gaussian Curvature of a Surface . . . . . . . . . . . . . . . 56
2.5.1 Riemann tensors . . . . . . . . . . . . . . . . . . . . 56
2.5.2 Gaussian curvature . . . . . . . . . . . . . . . . . . . 58
2.5.3 Geodesic curvature and normal curvature . . . . . . 59
2.5.4 Principal curvatures . . . . . . . . . . . . . . . . . . 60
2.6 Geodesic Equation . . . . . . . . . . . . . . . . . . . . . . . 63
2.7 Structure Equations in Differential Forms . . . . . . . . . . 64
2.7.1 Smooth surfaces in R3 and integrability . . . . . . . 64
2.7.2 Structure equations . . . . . . . . . . . . . . . . . . . 66
2.7.3 Geodesic equation. . . . . . . . . . . . . . . . . . . . 68
2.8 Gauss Spherical Map . . . . . . . . . . . . . . . . . . . . . . 69
2.9 Gauss–Bonnet Theorem I . . . . . . . . . . . . . . . . . . . 70
2.10Gauss–Bonnet Theorem II . . . . . . . . . . . . . . . . . . . 72
2.11Uniqueness: First and Second Fundamental Tensors . . . . 74
3. Riemannian Geometry 77
3.1 Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1.1 Tangent vectors and inner product . . . . . . . . . . 77
3.1.2 Riemannian metric . . . . . . . . . . . . . . . . . . . 78
3.1.3 Examples of metric tensor . . . . . . . . . . . . . . . 79
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3.2 Covariant Derivative (Connection) . . . . . . . . . . . . . . 80
3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2.2 Time-dependent case . . . . . . . . . . . . . . . . . . 81
3.3 Riemannian Connection . . . . . . . . . . . . . . . . . . . . 82
3.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3.2 Christoffel symbol. . . . . . . . . . . . . . . . . . . . 83
3.4 Covariant Derivative along a Curve . . . . . . . . . . . . . . 83
3.4.1 Derivative along a parameterized curve . . . . . . . . 83
3.4.2 Parallel translation . . . . . . . . . . . . . . . . . . . 84
3.4.3 Dynamical system of an invariant metric . . . . . . . 84
3.5 Structure Equations . . . . . . . . . . . . . . . . . . . . . . 85
3.5.1 Structure equations and connection forms . . . . . . 85
3.5.2 Two-dimensional surface M2 . . . . . . . . . . . . . 88
3.5.3 Example: Poincar´e surface (I) . . . . . . . . . . . . . 89
3.6 Geodesic Equation . . . . . . . . . . . . . . . . . . . . . . . 91
3.6.1 Local coordinate representation . . . . . . . . . . . . 91
3.6.2 Group-theoretic representation . . . . . . . . . . . . 92
3.6.3 Example: Poincar´e surface (II) . . . . . . . . . . . . 93
3.7 Covariant Derivative and Parallel Translation . . . . . . . . 96
3.7.1 Parallel translation again. . . . . . . . . . . . . . . . 96
3.7.2 Covariant derivative again . . . . . . . . . . . . . . . 98
3.7.3 A formula of covariant derivative . . . . . . . . . . . 98
3.8 Arc-Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.9 Curvature Tensor and Curvature Transformation . . . . . . 102
3.9.1 Curvature transformation . . . . . . . . . . . . . . . 102
3.9.2 Curvature tensor . . . . . . . . . . . . . . . . . . . . 103
3.9.3 Sectional curvature . . . . . . . . . . . . . . . . . . . 105
3.9.4 Ricci tensor and scalar curvature . . . . . . . . . . . 107
3.10Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . 108
3.10.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . 108
3.10.2 Initial behavior of Jacobi field . . . . . . . . . . . . . 111
3.10.3 Time-dependent problem . . . . . . . . . . . . . . . . 112
3.10.4 Two-dimensional problem . . . . . . . . . . . . . . . 113
3.10.5 Isotropic space . . . . . . . . . . . . . . . . . . . . . 114
3.11Differentiation of Tensors . . . . . . . . . . . . . . . . . . . 114
3.11.1 Lie derivatives of 1-form and metric tensor . . . . . . 114
3.11.2 Riemannian connection ∇ . . . . . . . . . . . . . . . 115
3.11.3 Covariant derivative of tensors. . . . . . . . . . . . . 116
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3.12Killing Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.12.1 Killing vector field X . . . . . . . . . . . . . . . . . . 117
3.12.2 Isometry . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.12.3 Positive curvature and simplified Jacobi equation . . 118
3.12.4 Conservation of (cid:2)X,T(cid:3) along γ(s) . . . . . . . . . . . 119
3.12.5 Killing tensor field . . . . . . . . . . . . . . . . . . . 120
3.13Induced Connection and Second Fundamental Form . . . . 121
II. Dynamical Systems
4. Free Rotation of a Rigid Body 127
4.1 Physical Background . . . . . . . . . . . . . . . . . . . . . . 127
4.1.1 Free rotation and Euler’s top . . . . . . . . . . . . . 127
4.1.2 Integrals of motion . . . . . . . . . . . . . . . . . . . 130
4.1.3 Lie–Poisson bracket and Hamilton’s equation . . . . 131
4.2 Transformations (Rotations) by SO(3) . . . . . . . . . . . . 131
4.2.1 Transformation of reference frames . . . . . . . . . . 132
4.2.2 Right-invariance and left-invariance . . . . . . . . . . 133
4.3 Commutator and Riemannian Metric . . . . . . . . . . . . . 135
4.4 Geodesic Equation . . . . . . . . . . . . . . . . . . . . . . . 137
4.4.1 Left-invariant dynamics . . . . . . . . . . . . . . . . 137
4.4.2 Right-invariant dynamics. . . . . . . . . . . . . . . . 138
4.5 Bi-Invariant Riemannian Metrices . . . . . . . . . . . . . . 139
4.5.1 SO(3) is compact . . . . . . . . . . . . . . . . . . . . 140
4.5.2 Ad-invariance and bi-invariant metrices. . . . . . . . 140
4.5.3 Connection and curvature tensor . . . . . . . . . . . 142
4.6 Rotating Top as a Bi-Invariant System . . . . . . . . . . . . 143
4.6.1 A spherical top (euclidean metric) . . . . . . . . . . 143
4.6.2 An asymmetrical top (Riemannian metric) . . . . . . 144
4.6.3 Symmetrical top and its stability . . . . . . . . . . . 146
4.6.4 Stability and instability of an asymmetrical top . . . 149
4.6.5 Supplementary notes to §4.6.3 . . . . . . . . . . . . . 150
5. Water Waves and KdV Equation 153
5.1 Physical Background: Long Waves in Shallow Water . . . . 154
5.2 Simple Diffeomorphic Flow . . . . . . . . . . . . . . . . . . 158
5.2.1 Commutator and metric of D(S1) . . . . . . . . . . . 158
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5.2.2 Geodesic equation on D(S1) . . . . . . . . . . . . . . 160
5.2.3 Sectional curvatures on D(S1) . . . . . . . . . . . . . 160
5.3 Central Extension of D(S1) . . . . . . . . . . . . . . . . . . 161
5.4 KdV Equation as a Geodesic Equation on Dˆ(S1) . . . . . . 161
5.5 Killing Field of KdV Equation . . . . . . . . . . . . . . . . 163
5.5.1 Killing equation . . . . . . . . . . . . . . . . . . . . . 163
5.5.2 Isometry group . . . . . . . . . . . . . . . . . . . . . 163
5.5.3 Integral invariant . . . . . . . . . . . . . . . . . . . . 164
5.5.4 Sectional curvature . . . . . . . . . . . . . . . . . . . 165
5.5.5 Conjugate point . . . . . . . . . . . . . . . . . . . . . 166
5.6 Sectional Curvatures of KdV System . . . . . . . . . . . . . 168
6. Hamiltonian Systems: Chaos, Integrability and Phase Transition 171
6.1 A Dynamical System with Self-Interaction . . . . . . . . . . 171
6.1.1 Hamiltonian and metric tensor . . . . . . . . . . . . 171
6.1.2 Geodesic equation. . . . . . . . . . . . . . . . . . . . 173
6.1.3 Jacobi equation . . . . . . . . . . . . . . . . . . . . . 173
6.1.4 Metric and covariant derivative . . . . . . . . . . . . 174
6.2 Two Degrees of Freedom . . . . . . . . . . . . . . . . . . . . 175
6.2.1 Potentials . . . . . . . . . . . . . . . . . . . . . . . . 175
6.2.2 Sectional curvature . . . . . . . . . . . . . . . . . . . 176
6.3 H´enon–Heiles Model and Chaos . . . . . . . . . . . . . . . . 177
6.3.1 Conventional method . . . . . . . . . . . . . . . . . . 177
6.3.2 Evidence of chaos in a geometrical aspect . . . . . . 177
6.4 Geometry and Chaos . . . . . . . . . . . . . . . . . . . . . . 178
6.5 Invariants in a Generalized Model. . . . . . . . . . . . . . . 181
6.5.1 Killing vector field . . . . . . . . . . . . . . . . . . . 181
6.5.2 Another integrable case . . . . . . . . . . . . . . . . 183
6.6 Topological Signature of Phase Transitions . . . . . . . . . 183
6.6.1 Morse function and Euler index . . . . . . . . . . . . 184
6.6.2 Signatures of phase transition . . . . . . . . . . . . . 185
6.6.3 Topological change in the mean-field XY model . . . 186
III. Flows of Ideal Fluids
7. Gauge Principle and Variational Formulation 191
7.1 Introduction: Fluid Flows and Field Theory . . . . . . . . . 191
7.2 Lagrangians and Variational Principle . . . . . . . . . . . . 193
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7.2.1 Galilei-invariant Lagrangian . . . . . . . . . . . . . . 193
7.2.2 Hamilton’s variational formulations . . . . . . . . . . 196
7.2.3 Lagrange’s equation . . . . . . . . . . . . . . . . . . 197
7.3 Conceptual Scenario of the Gauge Principle . . . . . . . . . 198
7.4 Global Gauge Transformation . . . . . . . . . . . . . . . . . 202
7.5 Local Gauge Transformation . . . . . . . . . . . . . . . . . 202
7.5.1 Covariant derivative . . . . . . . . . . . . . . . . . . 203
7.5.2 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 204
7.6 Symmetries of Flow Fields . . . . . . . . . . . . . . . . . . . 205
7.6.1 Translational transformation . . . . . . . . . . . . . . 206
7.6.2 Rotational transformation . . . . . . . . . . . . . . . 206
7.6.3 Relative displacement . . . . . . . . . . . . . . . . . 207
7.7 Laws of Translational Transformation . . . . . . . . . . . . 208
7.7.1 Local Galilei transformation . . . . . . . . . . . . . . 208
7.7.2 Determination of gauge field A . . . . . . . . . . . . 209
7.7.3 Irrotational fields ξ(x) and u(x) . . . . . . . . . . . . 210
7.8 Fluid Flows as Material Motion . . . . . . . . . . . . . . . . 212
7.8.1 Lagrangian particle representation . . . . . . . . . . 212
7.8.2 Lagrange derivative and Lie derivative . . . . . . . . 215
7.8.3 Kinematical constraint . . . . . . . . . . . . . . . . . 216
7.9 Gauge-Field Lagrangian L (Translational Symmetry) . . . 216
A
7.9.1 A possible form . . . . . . . . . . . . . . . . . . . . . 216
7.9.2 Lagrangian of background thermodynamic state . . . 217
7.10Hamilton’s Principle for Potential Flows . . . . . . . . . . . 218
7.10.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 218
7.10.2 Material variations: irrotational and isentropic . . . . 219
7.10.3 Constraints for variations . . . . . . . . . . . . . . . 220
7.10.4 Action principle for L . . . . . . . . . . . . . . . . . 221
P
7.11Rotational Transformations . . . . . . . . . . . . . . . . . . 224
7.11.1 Orthogonal transformation of velocity . . . . . . . . 224
7.11.2 Infinitesimal transformations. . . . . . . . . . . . . . 225
7.12Gauge Transformation (Rotation) . . . . . . . . . . . . . . . 226
7.12.1 Local gauge transformation . . . . . . . . . . . . . . 226
7.12.2 Covariant derivative ∇ . . . . . . . . . . . . . . . . 227
t
7.12.3 Gauge principle . . . . . . . . . . . . . . . . . . . . . 228
7.12.4 Transformation law of the gauge field Ω . . . . . . . 228
7.13Gauge-Field Lagrangian L (Rotational Symmetry) . . . . 229
B
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7.14Biot–Savart’s Law . . . . . . . . . . . . . . . . . . . . . . . 231
7.14.1 Vector potential of mass flux. . . . . . . . . . . . . . 231
7.14.2 Vorticity as a gauge field . . . . . . . . . . . . . . . . 233
7.15Hamilton’s Principle for an Ideal Fluid
(Rotational Flows) . . . . . . . . . . . . . . . . . . . . . . . 233
7.15.1 Constitutive conditions . . . . . . . . . . . . . . . . . 234
7.15.2 Lagrangian and its variations . . . . . . . . . . . . . 234
7.15.3 Material variation: rotational and isentropic . . . . . 235
7.15.4 Euler’s equation of motion . . . . . . . . . . . . . . . 237
7.15.5 Conservations of momentum and energy . . . . . . . 238
7.15.6 Noether’s theorem for rotations . . . . . . . . . . . . 240
7.16Local Symmetries in a-Space . . . . . . . . . . . . . . . . . 242
7.16.1 Equation of motion in a-space . . . . . . . . . . . . . 242
7.16.2 Vorticity equation and local rotation symmetry . . . 244
7.16.3 Vorticity equation in the x-space . . . . . . . . . . . 246
7.16.4 Kelvin’s circulation theorem . . . . . . . . . . . . . . 248
7.16.5 Lagrangian of the gauge field . . . . . . . . . . . . . 249
7.17Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
8. Volume-Preserving Flows of an Ideal Fluid 253
8.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . 254
8.1.1 Volume-preserving diffeomorphisms . . . . . . . . . . 254
8.1.2 Right-invariant fields . . . . . . . . . . . . . . . . . . 257
8.2 Basic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
8.2.1 Commutator . . . . . . . . . . . . . . . . . . . . . . . 260
8.2.2 Divergence-free connection . . . . . . . . . . . . . . . 261
8.2.3 Coadjoint action ad∗ . . . . . . . . . . . . . . . . . . 262
8.2.4 Formulas in R3 space . . . . . . . . . . . . . . . . . . 263
8.3 Geodesic Equation . . . . . . . . . . . . . . . . . . . . . . . 264
8.4 Jacobi Equation and Frozen Field. . . . . . . . . . . . . . . 266
8.5 Interpretation of Riemannian Curvature of Fluid Flows . . 268
8.5.1 Flat connection . . . . . . . . . . . . . . . . . . . . . 268
8.5.2 Pressure gradient as an agent yielding curvature. . . 269
8.5.3 Instability in Lagrangian particle sense . . . . . . . . 271
8.5.4 Time evolution of Jacobi field . . . . . . . . . . . . . 273
8.5.5 Stretching of line-elements . . . . . . . . . . . . . . . 273
8.6 Flows on a Cubic Space (Fourier Representation) . . . . . . 275
8.7 Lagrangian Instability of Parallel Shear Flows . . . . . . . . 277
8.7.1 Negative sectional curvatures . . . . . . . . . . . . . 277
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8.7.2 Stability of a plane Couette flow . . . . . . . . . . . 279
8.7.3 Other parallel shear flows . . . . . . . . . . . . . . . 284
8.8 Steady Flows and Beltrami Flows . . . . . . . . . . . . . . . 285
8.8.1 Steady flows . . . . . . . . . . . . . . . . . . . . . . . 285
8.8.2 A Beltrami flow . . . . . . . . . . . . . . . . . . . . . 287
8.8.3 ABC flow . . . . . . . . . . . . . . . . . . . . . . . . 289
8.9 Theorem: α1 =−i dα1 +df . . . . . . . . . . . . . . . . . 290
B u w
9. Motion of Vortex Filaments 293
9.1 A Vortex Filament . . . . . . . . . . . . . . . . . . . . . . . 294
9.2 Filament Equation . . . . . . . . . . . . . . . . . . . . . . . 297
9.3 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . 301
9.3.1 Left-invariance and right-invariance . . . . . . . . . . 301
9.3.2 Landau–Lifshitz equation . . . . . . . . . . . . . . . 302
9.3.3 Lie–Poisson bracket and Hamilton’s equation . . . . 302
9.3.4 Metric and loop algebra . . . . . . . . . . . . . . . . 304
9.4 Geometrical Formulation and Geodesic Equation . . . . . . 304
9.5 Vortex Filaments as a Bi-Invariant System . . . . . . . . . . 306
9.5.1 Circular vortex filaments . . . . . . . . . . . . . . . . 306
9.5.2 General vortex filaments . . . . . . . . . . . . . . . . 308
9.5.3 Integral invariants . . . . . . . . . . . . . . . . . . . 308
9.6 Killing Fields on Vortex Filaments . . . . . . . . . . . . . . 310
9.6.1 A rectilinear vortex . . . . . . . . . . . . . . . . . . . 310
9.6.2 A circular vortex . . . . . . . . . . . . . . . . . . . . 311
9.6.3 A helical vortex . . . . . . . . . . . . . . . . . . . . . 313
9.7 Sectional Curvature and Geodesic Stability . . . . . . . . . 314
9.7.1 Killing fields . . . . . . . . . . . . . . . . . . . . . . . 314
9.7.2 General tangent field X . . . . . . . . . . . . . . . . 315
9.8 Central Extension of the Algebra of Filament Motion. . . . 315
IV. Geometry of Integrable Systems
10. Geometric Interpretations of Sine–Gordon Equation 321
10.1Pseudosphere: A Geometric Derivation of SG . . . . . . . . 321
10.2Bianchi–Lie Transformation . . . . . . . . . . . . . . . . . . 324
10.3B¨acklund Transformation of SG Equation . . . . . . . . . . 326
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11. Integrable Surfaces: Riemannian Geometry and Group Theory 329
11.1Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
11.2Pseudospherical Surfaces: SG, KdV, mKdV, ShG . . . . . . 330
11.3Spherical Surfaces: NLS, SG, NSM . . . . . . . . . . . . . . 333
11.3.1 Nonlinear Schro¨dinger equation . . . . . . . . . . . . 333
11.3.2 Sine–Gordon equation revisited . . . . . . . . . . . . 335
11.3.3 Nonlinear sigma model and SG equation . . . . . . . 336
11.3.4 Spherical and pseudospherical surfaces . . . . . . . . 338
11.4B¨acklund Transformations Revisited . . . . . . . . . . . . . 340
11.4.1 A B¨acklund transformation . . . . . . . . . . . . . . 340
11.4.2 Self-B¨acklund transformation . . . . . . . . . . . . . 342
11.5Immersion of Integrable Surfaces on Lie Groups . . . . . . . 343
11.5.1 A surface Σ2 in R3 . . . . . . . . . . . . . . . . . . . 343
11.5.2 Surfaces on Lie groups and Lie algebras . . . . . . . 344
11.5.3 Nonlinear Schro¨dinger surfaces . . . . . . . . . . . . 346
11.6Mapping of Integrable Systems to Spherical Surfaces . . . . 348
Appendix A Topological Space and Mappings 351
A.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
A.2 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Appendix B Exterior Forms, Products and Differentials 353
B.1 Exterior Forms . . . . . . . . . . . . . . . . . . . . . . . . . 353
B.2 Exterior Products (Multiplications) . . . . . . . . . . . . . . 355
B.3 Exterior Differentiations . . . . . . . . . . . . . . . . . . . . 357
B.4 Interior Products and Cartan’s Formula . . . . . . . . . . . 358
B.5 Vector Analysis in R3 . . . . . . . . . . . . . . . . . . . . . 358
B.6 Volume Form and Its Lie Derivative . . . . . . . . . . . . . 361
B.7 Integration of Forms . . . . . . . . . . . . . . . . . . . . . . 362
B.7.1 Stokes’s theorem . . . . . . . . . . . . . . . . . . . . 362
B.7.2 Integral and pull-back . . . . . . . . . . . . . . . . . 363
Appendix C Lie Groups and Rotation Groups 365
C.1 Various Lie Groups . . . . . . . . . . . . . . . . . . . . . . . 365
C.2 One-Parameter Subgroup and Lie Algebra . . . . . . . . . . 366
C.3 Rotation Group SO(n) . . . . . . . . . . . . . . . . . . . . . 367
C.4 so(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
